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Unformatted text preview: 1. (50 points total)
Consider the following linear programming problem (P) maximize 30ml + 16322 + 363:3 + 15234 subject to: 23:1 + 1232 —i— 2333 + 1234 S 28
4331 + 2.2132 + 5$3 + 2334 S 60
1331 + 2232 + 8333 + 0394 g 72 $1 2 0:32 201553 2 01174 2 0 in which we are maximizing proﬁt in thousands of dollars) from the production of four
different products that com etc for three scarce resources. We have added slack vari
ables 935, $6, $7 in the ﬁrst, second, and third functional constraints, respectively, to get
a tableau, Tableau I (not shown), corresponding to the ordered basis (5, 6, 7). After per
forming several simplex iterations we have reached the following optimal tableau, which we will call Tableau II. Tableau II Z 531 $2 $3 $4 $5 $6 {137 (a) For Tableau II identify:
i (2 points) The ordered basis B and the objective function value. \A b l? 0J4“? C13 (72 3) FJZ'LEQEJCD O“ U” a.  .. .... _._. "1.53 \ cﬁ‘e 8 Pt ii. (6 points) (3,143 and A517" (b) (8 points) There is a concern that 55,55,57 and (—23 may be wrong. Use A; to
calculate the vector 7r of marginal prices and to verify that the entries 55, 55, E7 and
b3 are correct (make sure to ShOW your work).  \
f1; Q :(gﬁp) I Q L»%”bli® ”Q'V‘CCCI‘M W Tableau II Z :61 :62 333 $4 $5 $5 .717 Consider the remaining‘parts (c),(d), (ejand (f) independently. (c) (5 points) Suppose there is an open market for resource 1 (with slack variable :55).
Explain brieﬂy, but care‘ﬁgllyhhowyou would determine this ﬁrm’s breakeven market price for resource 1 .9»— _ ' .5»: PM"
i l\ 4r” 3' a \f3\nﬁfe Ell \g All“ awfﬁl 0 LL Sula 3 3' (2': (343)? :9} We gr; u 'r en. l l r; , AQ’lQ Awlmjﬂns mew«~31 =
”Mun—trun— swank7n :_:ra.w,ir,.;«a “are, __~rp'L‘:1._\:;'_‘ ‘ usm‘ '. .   a  . .w . ”"7 7 l h  A A. . ‘ . F1 4,.
CA ) an 5 4:13 1 (3' H”  T? J: : ”it; {we WW1“ ’t 3”” U“ i S WKWMEWLK. (d) i. (3 points) A new product, Product 0, is under consideration. The “recipe” for
this product is A0 = (9, 20, 6)T. For what range of co, the unit proﬁt for Product
0, would it be advantageous to produce some Product 0'? Explain. In 1: ~m“ m A l ACEUQKM'RE eve9 war F‘l' Haiti? M7 gel“? 1”? AG
(El , < ,,C;c> "rid a ' 0 ii. (4 points) Assume that (:0 is in the range you speciﬁed in part (i), so we want to
use the simplex method to pivot are into the old optimal basis. Determine the
leaving variable. (You do not need to update the tableau.) Tableau Il Z 331 $2 $3 $4 $5 $6 $137 (e) (18 points total) Suppose that the availability of the second scarce resource is ﬂuc
tuating. Let [)2 = 60 + 6. (5 points) (i) Find the new values of the basic variables and the objective function correspond
ing to the previously O’BFl/Hjlal basisB from part (a(i)) in‘terms of 6. m _,____,_..._.rs. —=~ l A”
P r ' M “Wl ”cl
A) F; ‘ , “:5 t: a“ 1, Q l (5:294 C: E
”NEW“... l” 2 1 a L 71
q T" ,ﬁ_:'“llll!€3*=ilg
"a All i? i” A; \ it t: P {Q “Li ll PPPLE"
.1: o} Li}x\\~k{,,3;
,. .y » r ,i i“
ﬁrsml “241 *1 ”5 Eu; ,1 I '1. T 3 U: LQ’JEXF: L“ w \@W
W :2; m {w H, 2 iv (3 points) (ii) Find the range of values of 52 suchthat Ithelig‘ljahG‘iOusly optimal basis B from
part (3(1)) remains optimal. _ \ WE’EQ lisl, “leﬁfig Clsfl 65$? (:3 "3 El ‘5: “’{l (10 points) (iii) Suppose now that (72 = 60 + E for E > 0. Do one dual simplex pivot starting
from the previously optimal basis B from part (a(i)), updating only the solution
column and objective function row, and determine the range of values for e such that the new basis is optimal. (f) (4 points) Suppose we need to add the new constraint: 321 + 2E2 + .133 S 40. Let (as
be the slack variable that is added to the new constraint. You wish to use the dual
simplex method to re—optimize subject to the additional constraint starting from the
tableau, Tableau III, associated with ordered basis (BI, .82, BB, 8), where (51,32, .83)
is the 01d optimal basis from part (a{i)). Derive the $3  row of Tableau III. (Do not do any pivoting). i (iii “ i Y: \s; e o a. sinwww day—v V
71
ET Cl bgwﬁ_aﬁfx Leila1, 2. (50 points total) During the upcoming production period MBC (MoveyourBrass Corp.)
will produce several alloys from different combinations of metals. The prices and avail—
abilities of the metals are known (MBC has contracts with a supplier to buy up to the
known availability at the given price.) In addition to the cost of the metals, MBC incurs
an additional production cost per ton of alloy (the same known additional production
cost for all alloys). MBC can sell everything it produces; the revenue per ton for each
alloy is known. Below is an AMPL model ﬁle for optimizing production of the alloys.
The model ﬁle is not complete, you will need to ﬁll in some details later. set INPUTS; # inputs
set OUTPUTS; # outputs param availability {INPUTS} >= 0; in tons param price {INPUTS} >= 0; per ton param prodcharge >=0; # additional production price per ton
# same for all OUTPUTS param revenue {OUTPUTS} >= 0; per ton param min_ limit {INPUTS OUTPUTS} >= 0; param max_ limit {INPUTS OUTPUTS} __?; smwmw‘) if‘ when a m L’W‘P‘ﬂ
., in _2a5\(r5; l ml1mNA—‘l' ,u
yamsf: ” ﬁﬁ; ‘30? \VQ maximize tot.al_ rofitI “ ' ' " ._ .; I I' f
D Ii!) P g K, ' ll'NlljlJ Y1“ A 1.“ QJI life} 7‘
’:th Elli? .‘r i. m .it “3‘“ 15M 6‘15 i (“a if?) rd. “warmsW" MM 7.2 m
\WW" _; huge«F Lu? subject to Scarcity {i in INPUTS}:
sum {j in OUTPUTS} X[i,j] <= availabilityIi]; subject to min_content_requirements {i in INPUTS, j in OUTPUTS}: subject to max__ contentd requirements {i in INPUTS, j in OUTPUTS}.
max_limit[i, j] * sum {it in INPUTS} XEk, j] >= XEi, j] ; (a) (3 points) Complete the variable declaration in the space given above in the model
ﬁle. (b) (5 points) Complete the objective function in the space given above in the model
ﬁle. In each of parts (c)(d) give a brief but precise interpretation of the speciﬁed statement
in the model ﬁle. If the statement is indexed over one or more sets, your interpretation should reﬂect that (for example, the interpretation of the scarcity constraint 1n part (c) might begin with “for each ...” to reﬂect that the constraint is indexed. Include u111ts of measurement Where appropriate. .
. i1)
4 pomts g. In one Lsentence inter renthi scarcit constraint. ”
W’” I) y ' (ML (r {is Hm l?“
\jlw s r @538 Pale; ...l.. :‘1 Visit“ Wisso issﬁ'ig 3, Reg; wise. V(L H imqueg o._ :cim) i i
,i i 1 .7 is t i. (is we) #6?)
re: mihwg ”Ham5r ’lylM Vii: “Ms,“ QM»— V‘hkswmj , was D ,,
1*" B, I. I
at b \ 1n owl? e» U "‘ “M 3 Ugﬁél :l in E Gill Hid V“ i ii to“, c} we WA ‘1“ “s“ W s . ii. iii. ,4 j
\ g, (4 points) ((21) In one sentence interpret the max_ content _requirements constraint as clearly and
simply as you can. (Hint: do not simply compare the right hand— side with the left— handside. ) w . f Qﬁll (1% 15:34 IiiVA“ i1! \egor Judi/n" [Liomhlrsa Keir} oil. a?“ “a?“ ‘l‘. M ,_ ,.
I ’ A,.. if.»
as. .Qiowitkf‘amt (‘35; ””1“ 1 \e " : \MEGEOE .. ,_ .. i .,.,..,,i_ ..
saw 3 O? . WM ’57 e... e‘tu. mice : gar su (e) (19 points total} You have not seen the data ﬁle, which, among other things, gives
revenues and prices in dollars per metric ton and availabilities in metric tons. On the next page you will ﬁnd a partial output ﬁle. ampl: solve;
CPLEX 11.2.0: optimal solution; objective 4967500 7 dual simplex iterations (6 in phase I)
X := Copper Alloyl 75
Copper Alloy2 125
Copper Alloys 0
Tin Alloyi 125
Tin Alloy2 75
Tin AlloyS O
Zinc Alloyi 50
Zinc Alloy2 50
Zinc AlloyB 0 c
2 Scarcity [*J Copper 11958.8
Tin 12878.7
Zinc O 5 (3 points) (ii) For each of the input rectals write down (as a. sum, if you like), how much of 1t MBC will purchase this period. __.... . v .2 5;.
O) (13:: ppm} ‘, ”7 5; 4” \ "2.3? .3 we "“2 QC) D ‘43:?“ 7‘ mm W” ﬁr C: ’1‘, an“; is; n27— (:5 :5: '62:. C2: {:1 WVEV‘IE {El,1 1 i V.) t E I _ h LR“€; 11"} C‘ 13’ Jr 5’ 0+") m“: 1 Q 1" "3 {1.5 u .. . _ . o .23 — . . M M _
11 \ Era \ﬂ ‘3; , _,,. ,1;  k (3 points) (iii) For each of the OUTPUTS write down (as a sum, if you like), how much of it MBC will produce this period.
:1 5?; +5ng (9 #Hﬁa“ ; ﬁgs 12.5“ 45:5 ,2:
= m ':: ts 3“ «.2 mg
o mum: new 155% its w '
V “:2 C) “timer:
Jr C) 3 .3
xi) AHQE D 6 MM
.1... 6? 1‘ 10 (6 points) (iv) Give as much information as you can about the data ﬁle statementspecifying W :71
the values of the parameter availability. Explain carefully. '1? {ma slats}? (“l 5119.62? ('4 11":
“new W...) H3 _. " 7..“ We NM l‘ﬁ M02: (4 points) (v) There is a market for copper in which MBC can sell some of the copper available ”W 14‘s ‘13 2/ to it, or buy some additional copper. Give as much information as you can about
MBC’s breakeven price for copper in this market. .
(1.? El. l ”l’fol 7: . ’ i" l f“  aé’id’
W‘xiir‘r Wall. bf" GE“! “* 45" UM“ lbl‘ “if"  H gm); l): \ \ q $6133: ”:59 a", Pr ; C, a; 536??“3‘11; The remaining two parts (f) and (g) should be done independently. (f) The production program from the optimal solution given above has been rejected by
the plant manager. After some conversations with her, you have added the following
two statements to the model ﬁle: # newr
param new_rhs ;
# #new
subject to New_constraint:
sum {1 in INPUTS, j in OUTPUTS} X[i,j] <= new_rhs; # (4 points) (i) Give a non—technical interpretation of the new constraint. Be as explicit as possrble. , ' 1 l‘ . , , as. {be.4
l m {\JCI ‘3'th DAV.“ $1 _,_ can 3"? {a l“? “WW”W‘7;NWWWVSWMy”;,.,__. 7 :1 a.
,1 luau “ﬁg” 1 ”m l ' . (1‘ ‘ﬁ irons) jr'l «— ‘le:  9' I é‘ﬁ ;
WW ”VFW: 33H \ M
' ' ' h ' l. Lane a .A r~ ,
9k “Flam haemog , ,_ W W a m: ll (3 points)(ii) Give as much information as you can on how the data ﬁle should be changed to
accommodate the new constraint. Your response should include information on values in any new data ﬁle statements. .‘WS gal , l""§".'1’/Gﬁ 3%;“EJ: Jl’lw olgl GFV\W¢‘1L\N
3 *‘l “I" °"“ 59“” m lLEZCELMTQST; ssh .ala’msWs mtmm um...“ . n" ﬁt: a (g) After Consultatﬁin with big ‘erups, the plant manager has a new version of the model
ﬁle, which begins as follows: set METALS ; set SOURCES {METALS};
set INPUTS = union {i in METALS} SOURCES [i]; # inputs set OUTPUTS; # outputs param availability {INPUTS} >= 0; in tons param price {INPUTS} >= 0; per ton
param prodcharge >=0; # additional production price per ton
# same for all OUTPUTS param revenue {OUTPUTS} >=0; per ton
param min_limit {METALS,OUTPUTS} >= 0;
param max_limit {METALS,OUTPUTS] >= 0; When the complete model ﬁle and an appropriate data ﬁle for this version are run
in AMPL, we get the following output (see the next page): 12 CPLEX 11.2.0: optimal solution; objective 6693250
13 dual simplex iterations (11 in phase I) X := Copperl Alloyi 12.5
Copperl Alloy2 187.5
Copperi AlloyS O
Copper2 Alloyi _100
Copper2 A110y2 O
Copper2 Alloys 0
Tinl Alloyi 187.5
Tini Alloy2 12.5
Tini AlloyB O
Tin2 Alloyi O
Tin2 Alloy2 100
Tin2 Alloy3 0
Zincl Alloyl 75
Zinci Allon 75
Zinci AlloyS 0
Zinc2 Alloyi O
Zinc2 AlloyZ 0
Zinc2 Alloy3 0 J Scarcity [*J := Copperl 11958.75 Copper2 9878.75
Tini 12878.75
Tin2 7878.75
Zincl 0
Zinc2 O (6 points) (1) Describe the scenario behind this model. Be brief but as precise as possible.
Include, where possible, information on the optimal production program, avail— abilities of resources, and ”relative” prices,
“i’iocwx \‘V’Vi “gt '51 ED \ (Vie. $23 v $111. “with r” G I‘fArLﬁ—M” (:3 W5! AW; 5.4" r1 2 § at”; “M3
«.4 wk We W i __ °‘ 4 +4++4+
“1414mm; ”1:4444 ++..........4++.++»++4+++++4+z L W” MS mtg) have ch99; mm D
" 7’! \. ’\" 2— S 0"“ r ‘r '''' ‘
So ( I: no. 0 r“ “ 2::
. . 4/ \ o
Clm‘ \oiﬁidixhﬂw DANA “95...? ‘ ii». CNQAGLM ‘5
(”1.1.144 . .3 W...“  \" ‘1 1 em "in ”/7 {\
CPR?! ”Sr Q’W '2, 1 e o i “‘6' A 24»;
l j!
'7 Tit“ {swim 3' LOU :{tjsﬂf/
i ‘ 3MWUYL‘ Hiiob “V ~,,73“a
1:“ * 1 $1“; \ i? We: it “1 \ﬁ/i? ’3
s v:  _
(7:741: ‘5 1310”"; he“ 7 CF” “CL. ..Ti79"‘~5" V "I S m x} f C». 7» (33*?VEA U“ “A“ “3‘” "M W "2;,“ \ , bx \ﬂNmﬁa MEWS'NK ‘W E. bwﬂ I 13 9 9 e r Nb , J/C‘ugﬁ C393? 2 (j Qwé wM‘n) Fee: < ’38“ N12 \ ii“ _
“‘1' A53“: "“"‘“~'—“Y"“"t'f(;:1w;=risking,“ “S, 6” .151,» .217. L::." .¢')¥x'r.K , a“ _.L, .. . W— “why“ lr‘m (3 points)(ii) Explam why 1n the output from the AMPL command
display Scarcity
the value for
Tinl
is greater than the value for
Tin2
by $5,500 per ton. ...
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